Determining D traveled by xForce. (notschoolwork)

Not schoolwork but a question we have been looking at for awhile at work.

1. The problem statement, all variables and given/known data
Basically we want to know if I can push a rectangular toolbox thing off of 1 counter and have it land on another.

The box is 4kg
The height difference between the 2 shelves is .5m
Distance between the 2 is lets say 2m

2. Relevant equations

a = 9.8m/sec/sec
mass = 4kg
f = m x a
d = 2m

So basically I need to know how to determine the trajectory of the object at x force so I can determine how much force I need to use to move it to the desired final position. If I could find a formula that would let me say use 1newton(i think thats what force is in) of force against it I could see where it would be after 2m of distance then just continue up until I reach the force I need to make it land on the other side (if its possible for me to even exert that much force.

3. The attempt at a solution

kinda tried on some paper but I dont know how to get the formula to figure out the trajectory because the object will be moving forward and dropping at the same time at different rates depending on the force and such. Any help? Im more then happy to spend some time working on it I just dont know where to start!

Actually it was, and I broke it cause its plastic but duct tape is amazing and I fixed it. Sorta... but I "almost" made it, had I took the fact that the front half was lighter then the second half of the box and that it would and up hitting like /- instead of -_ like I hoped then I would have had it. But now that we all accepted that its possible I still want to figure out the equation for the amount of force needed to make it work.

You should start by calculating the initial velocity the toolbox should have when it just been pushed off the counter. I assume that the force applied is going to be horizontal only, so that means to say horizontal velocity of the toolbox in the air would be constant. You have the equations:

s=ut +1/2at^2
v^2 = u^2 + 2as
v = u+at

So with these you should be able to figure out the initial velocity of the toolbox. I'm ignoring factors such as drag in the air just to simplify matters here. Calculating the force needed is more difficult because you need to know the coefficient of dynamic and static friction as well as consider over what distance on the counter you would be pushing the box.